Abstract

In psychological test norming, one needs to estimate the complete distribution of test scores conditional upon covariate(s). Also in growth chart estimation, the conditional distribution is of key interest. In test norming, normed scores (e.g., IQ-scores, percentiles, Z-scores) indicate the test taker's relative position compared to a reference population, like the general population of the same age. Herewith, the tails of the conditional test score distribution are of particular interest, because critical life decisions tend to concern ‘extreme’ cases (e.g., diagnosis of a psychological disorder based on a validated questionnaire, and admission to an elite university based on an admission test).
The (conditional) distribution can be modeled in different ways. We focus on parametric and nonparametric approaches. The nonparametric approach, unlike the parametric approach, does not require the –sometimes burdensome – selection of a specific test score distribution, and provides direct insight into the relationships between the covariates and the ‘extremeness’ of the distribution. In contrast, the parametric approach may be more efficient and provides direct insight into the relationships between the covariates and the distributional parameters (as the center, spread and skewness).
As the family of parametric models, we consider the generalized additive models for location, scale and shape (GAMLSS). As the nonparametric models, we consider quantile and expectile regression, in particular the classical curve variant and the sheet model. Unlike quantile regression, as far as we know, expectile regression has not been proposed in the norming context. Expectile regression can be expected to be more efficient than quantile regression for distributions close to Gaussian, and for distributions with low densities in the tails. Yet, expectile regression seems to be more sensitive to outliers. In the curve model, the p-th quantile/expectile curve is estimated as a function of the covariate, with p being the quantile/asymmetry parameter. In the sheet model, the surface of the covariate and p is estimated, providing a complete model. As the sheet model, unlike the curve model, properly expresses the underlying continuity of quantile/asymmetry parameters, it can be expected to be more efficient than the curve model.
We will compare the performance of these parametric and nonparametric models for estimating the conditional test score distribution, in terms of efficiency, with a particular focus on the tails. Further, we will investigate the robustness of GAMLSS models against model misspecification. We will do so using a simulation study, covering a broad range of conditions that are relevant from a theoretical and/or applied point of view. We will demonstrate the practical relevance of the findings for psychological test norming, using normative data from a published intelligence test and an instrument to screen for psychosocial difficulties.